1. Field of the Invention
The present invention relates generally to steam turbine rotor blades and, more particularly, to a retrofitted rotor blade for a pre-existing steam turbine and method of designing the same.
2. Description of the Related Art
Steam turbine rotor blades are arranged in a plurality of rows or stages. The rotor blades of a given row are identical to each other and mounted in a mounting groove provided in the turbine rotor.
Turbine rotor blades typically share the same basic shape. Each has a root receivable in the mounting groove of the rotor, a platform which overlies the outer surface of the rotor at the upper terminus of the root, and an airfoil which extends upwardly from the platform.
The airfoils of most steam turbine rotor blades include a leading edge, a trailing edge, a concave surface, a convex surface, and a tip at the distal end opposite the root. The airfoil shape common to a particular row of rotor blades differs from the airfoil shape for every other row within a particular turbine. Likewise, no two turbines of different designs share airfoils of the same shape. The structural differences in airfoil shape result in significant variations in aerodynamic characteristics, stress patterns, operating temperature, and natural frequency of the airfoil. These variations, in turn, determine the operating life of the rotor blades within the boundary conditions (turbine inlet temperature, compressor pressure ratio, and engine speed), which are generally determined prior to air foil shape development.
Development of a turbine section for a new commercial, power generation steam turbine may require several years to complete. When designing rotor blades for a new steam turbine, a profile developer is given a certain flow field with which to work. The flow field is determined by the inlet and outlet angles (for steam passing between adjacent rotor blades of a row), gauging, and the velocity ratio, among other things. "Gauging" is the ratio of throat to pitch; "throat" is the straight line distance between the trailing edge of one rotor blade and the vacuum-side surface of an adjacent blade, and "pitch" is the distance between the trailing edges of adjacent rotor blades.
These flow field parameters are dependent on a number of factors, including the length of the rotor blades of a particular row. The length of the blades is established early in the design stages of the steam turbine and is essentially a function of the overall designed power output of the steam turbines and the power output for that particular stage.
An essential aspect of rotor blade design is the "tuning" of the resonant frequency of the rotor blade so as to avoid resonant frequencies which coincide with or approximate the harmonics of running speed. Such coincidence causes the blades to vibrate in resonance, thereby leading to blade failure. Therefore, in the process of designing and fabricating turbine rotor blades, it is critically important to tune the resonant frequencies of the blades to minimize forced or resonant vibration. To do this, the blades must be tuned to avoid the "harmonics of running speed".
The harmonics of running speed is best explained by example. In a typical fossil fuel powered steam turbine, the rotor rotates at 3,600 revolutions per minute (rpm), or 60 "cycles" per second (cps). Since one cps equals 1 hertz (Hz), and since simple harmonic motion can be described in terms of the angular frequency of circular motion, the running speed of 60 cps produces a first harmonic of 60 Hz, a second harmonic of 120 Hz, a third harmonic of 180 Hz, a fourth harmonic of 240 Hz, etc. Blade designers typically consider frequencies up to the seventh harmonic (420 Hz). The harmonic series of frequencies, occurring at intervals of 60 Hz, represents the characteristic frequencies of the normal modes of vibration of an exciting force acting upon the rotor blades. If the natural frequencies of oscillation of the rotor blades coincide with the frequencies of the harmonic series, or harmonics of running speed, a destructive resonance can result at one or more of the harmonic frequencies.
Given that exciting forces can occur at a series of frequencies, a blade designer must ensure that the natural resonant frequencies of the blades do not fall on or near any of the frequencies of the harmonic series. This would be an easier task if rotor blades were susceptible to vibration in only one direction. However, a rotor blade is susceptible to vibration in potentially an infinite number of directions. Each direction of vibration will have a different corresponding natural resonant frequency. The multi-directional nature of blade vibration is referred to as the "modes of vibration". For a row of lashed rotor blades, up to at least seven different modes or directions of vibration are considered. Each mode of vibration establishes a different natural resonant frequency for a given rotor blade for a given direction.
The first mode of vibration is a tangential vibration in the rotational direction of the rotor, and is substantially influenced by the position of the lower of the two lashing wires used to interconnect a group of rotor blades. Lowering the position of the lower lashing wire tends to increase the resonant frequency for the first mode of vibration. The second mode of vibration is a tangential vibration in the axial direction of the rotor. The position of the lower lashing wire tends to have an inverse effect on the second mode frequency such that, as the lower wire is lowered to raise the frequency in the first mode, the frequency of the second mode falls. The third mode of vibration is vibration in the "X" direction such that displacement occurs in the axial direction of a wired group of blades. The third mode of vibration is highly dependent on the number of blades per group; the frequency is lowered with the addition of more blades in the group. As an example, viewing three blades lashed together in a group from the top, a third mode vibration would involve displacement of the outer two blades in opposite directions from the axial line of the three blades. The middle blade would have zero displacement. As the blades vibrate, the outer two blades reverse displacement in a vibratory fashion. In this respect, the third mode of vibration is a twisting or torsional type of vibration. The fourth mode of vibration is an in-phase vibration which is highly dependant upon on the positioning of the outer-most lashing wire; moving the outer-most lashing wire downwardly lowers the frequency in the fourth mode.
Modes of vibration beyond the fourth mode become increasingly complex. The fifth mode is considered a second "X" direction vibration, while the sixth mode is in a "U" direction, such that in the example of three blades lashed together, the U-directional vibration would involve displacement of the outer blades in the same direction and displacement of the center blade in the opposite direction. The seventh mode of vibration is another in-phase vibration.
When tuning lashed rotor blades, it is important to tune the blades with respect to the first three modes of vibration. Keeping in mind the harmonic series described above for a fossil fuel powered steam turbine operating at 3,600 rpm the natural resonant frequency for a rotor blade must be tuned to avoid frequencies at intervals of 60 Hz. For example, the second harmonic occurs at 120 Hz and the third harmonic occurs at 180 Hz. The standard practice is to attempt to tune the blade having a frequency falling somewhere between 120-180 Hz to come as close as possible to the mid point between the two harmonics, i.e., 150 Hz. It is not unusual to have a rotor blade having a natural resonant frequency which falls between the second and third harmonics for the first mode of vibration. Therefore, it is desireable to tune the blade to have a frequency at or near 150 Hz for the first mode of operation.
Frequencies for the second and third modes of vibration are similarly tuned to be as close as possible to a midpoint between two successive harmonics. However, frequency tests are commonly run up to and beyond the seventh mode of vibration. With respect to the fourth mode of vibration, a frequency near the seventh harmonic (420 Hz) might be expected; therefore, the outer-most lashing wire should be positioned to make sure that the resonant frequency for the fourth mode of vibration is sufficiently above the seventh harmonic.
When a new steam turbine is designed, the blade designer must tune the turbine blades so that none of the resonant frequencies for any of the modes of vibration coincide with the frequencies associated with the harmonics of running speed. Sometimes, tuning requires a trade off with turbine performance or efficiency. For instance, certain design changes may have to be made to the blade to achieve a desired resonant frequency in a particular mode. This may necessitate an undesirable change elsewhere in the turbine such as a change in the velocity ratio or a change in the pitch and width of the blade root.
A difficult problem arises in the situation where a pre-existing turbine is upgraded to increase its power output. This may be done by increasing the length of the blades of one or more rows, and boring out the cylinder around the row to accommodate the greater overall length. The new, longer blade would have to have substantially the same root to avoid having to replace the rotor. As a result of blade lengthening, such as from a 25 inch blade to a 26 inch blade, the originally designed and meticulously calculated flow field changes so that a redesign of the longer blade is necessary. This is more difficult than designing an original blade since the pre-existing turbine establishes non-variable or restricted design parameters. For instance, for every radial cross-section passing through two adjacent rotor blades, the concave surface of one of the rotor blades must converge with the convex, opposing surface of the other rotor blade, with the convergence being from the leading edge to the trailing edge. This must be done while maintaining a velocity ratio at or below a certain level. Also, as previously mentioned, the root cannot be altered.
Another particularly difficult problem with a retrofitted, longer blade for a pre-existing turbine is that, while the harmonics of running speed do not change, the natural resonant frequencies for all modes of vibration are decreased as a result of the blade lengthening. The resonant frequencies must be increased by tuning techniques which do not hamper performance or efficiency to an unacceptable degree.